reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem Th35:
  for T being homogeneous non empty TopSpace st ex p being Point
  of T st for A being Subset of T st A is open & p in A holds ex B being Subset
  of T st p in B & B is open & Cl B c= A holds T is regular
proof
  let T be homogeneous non empty TopSpace;
  given p being Point of T such that
A1: for A being Subset of T st A is open & p in A holds ex B being
  Subset of T st p in B & B is open & Cl B c= A;
A2: [#]T <> {};
  now
    let A be open Subset of T, q be Point of T such that
A3: q in A;
    consider f being Homeomorphism of T such that
A4: f.p = q by Def6;
A5: f"A is open by A2,TOPS_2:43;
    reconsider g = f as Function;
A6: dom f = the carrier of T by FUNCT_2:def 1;
A7: rng f = [#]T by TOPS_2:def 5;
    then g".q = f".q & g.(g".q) in A by A3,FUNCT_1:32;
    then
A8: g".q in g"A by A6,FUNCT_1:def 7;
    p = g".q by A4,A6,FUNCT_1:32;
    then consider B being Subset of T such that
A9: p in B and
A10: B is open and
A11: Cl B c= f"A by A1,A8,A5;
    reconsider fB = f.:B as open Subset of T by A10,Th24;
    take fB;
    thus q in fB by A4,A6,A9,FUNCT_1:def 6;
A12: f.:Cl B = Cl fB by TOPS_2:60;
    f.:Cl B c= f.:(f"A) by A11,RELAT_1:123;
    hence Cl fB c= A by A7,A12,FUNCT_1:77;
  end;
  hence thesis by URYSOHN1:21;
end;
